theorem
  for H being strict Subgroup of G holds Left_Cosets H = the set of all {a}
    implies H = (1).G
proof
  let H be strict Subgroup of G;
  assume
A1: Left_Cosets H = the set of all {a};
A2: the carrier of H c= {1_G}
  proof
    set a = the Element of G;
    let x be object;
    assume x in the carrier of H;
    then reconsider h = x as Element of H;
A3: h in H;
    reconsider h as Element of G by Th42;
    a * H in Left_Cosets H by Def15;
    then consider b such that
A4: a * H = {b} by A1;
    a * h in a * H by A3,Th103;
    then
A5: a * h = b by A4,TARSKI:def 1;
    1_G in H by Th46;
    then a * 1_G in a * H by Th103;
    then a * 1_G = b by A4,TARSKI:def 1;
    then h = 1_G by A5,GROUP_1:6;
    hence thesis by TARSKI:def 1;
  end;
  1_G in H by Th46;
  then 1_G in the carrier of H;
  then {1_G} c= the carrier of H by ZFMISC_1:31;
  then {1_G} = the carrier of H by A2;
  hence thesis by Def7;
end;
